Appl. Phys. A 68, 691–697 (1999) / DOI 10.1007/s003399900049 Applied Physics AMaterialsScience & Processing Springer-Verlag 1999

Observation of zero creep in piezoelectric actuatorsK.R. Koops1,∗, P.M.L.O. Scholte1, W.L. de Koning2

1Delft Institute of Microelectronics and Submicrontechnology, Faculty of Applied Physics, Delft University of Technology, Lorentzweg 1,2628 CJ, Delft, The Netherlands(Fax: +31-15/278-3251)2Mathematical System Theory Group, Faculty of Information Technology and Systems, Delft University of Technology, Delft, The Netherlands

Received: 15 February 1999/Accepted: 16 February 1999/Published online: 28 April 1999

Abstract. Piezoelectric actuators are frequently used nowa-days in a wide variety of positioning devices. Although verysuitable for small displacements in the range ofnm to severalhundreds ofµm, the actuators always suffer from hysteresisand creep between the input voltage and resulting displace-ments. In scanning applications, the input voltage is oftenused as an indicator of the induced displacement. This pro-cedure can result in a large position error depending on theamount of hysteresis and creep. In order to describe and con-trol hysteretic systems various models for hysteresis havebeen published but little is known about relaxation and creepin piezo materials. In this paper we present detailed studiesof the hysteretic behavior and piezo relaxation and creep. Wehave identified certain locations on the hysteresis loop thatexhibit zero creep. From this observation, a more fundamen-tal relation between the amount of creep and the local slopeof the hysteresis loop and the virgin curve is presented. Thisobservation could be useful in both open-loop and closed-loop position control, since it allows quantification of thecreep. Futhermore, the experimentally observed relation be-tween the creep and the hysteresis suggests a reduction ofthe creep for non-hysteretic transfers. First measurements ona system with reduced hysteresis support this hypothesis.

PACS: 77; 07.79.v; 62.20.hg

In our effort to design and construct a sub-angstrom reso-lution positioning device for scanning probe applications, wehave made extensive study of piezo hysteresis, relaxation, andcreep. The main goal of the current project is to combineprecision mechanics, sensor technology, and digital signalprocessing into a 3D translation stage with positioning andtracking capabilities beyond the angstrom level. Straightfor-ward applications will be in the field of metrology, nano-lithography and atomic manipulation. An operational proto-type is currently used to perform pilot experiments in orderto increase our understanding of piezo behavior and non-linear control applications. Since the piezoelectric actuator

∗ E-mail: r.koops@tn.tudelft.nl

is the main source of non-linear and hysteretic behavior ofthe translation device we have developed a mathematical de-scription for the piezo hysteresis, adapted from the field offerromagnetic hysteresis, to be used in a model-based dig-ital controller. In previous studies it has been shown thatthis model provides a satisfactory description for both mag-netic [1] and piezoelectric hysteresis [2] in the limit of smallinput signals, i.e. far from the region of saturation. How-ever, relaxation effects are not incorporated in this model. Inorder to be able to augment the model, sufficient informa-tion about relaxation effects in piezoelectric materials has tobe obtained. Furthermore, the experimental conditions in thestudy of ferromagnetic materials can be devised such that theso-called anhysteretic curve is formed [3, 4]. As the name im-plies, this curve in theB–H plane does not exhibit hysteresis(although still non-linear). The experimental conditions in-volve the application of a relaxation signal in the form of anac magnetic field once a static field is applied. An equiva-lent curve for piezoelectric materials would offer substantialimprovement of the positioning capabilities since only theremaining non-linearity has to be compensated. This paperwill describe the experimental set-up and measurement pro-cedures to perform relaxation measurements in piezoelectricmaterials. The results of these measurements have induceda more specific study of piezo creep, a phenomenon that canbe considered as piezo relaxation in the absence of an ex-ternally applied relaxation signal. The results from our creepexperiments have, finally, led to the study of creep in a systemwith reduced hysteresis.

1 Experimental details

The measurements were performed on a home-built piezo-actuated STM positioning device [5]. This device is equippedwith capacitive position sensors for both theX andY trans-lation directions and serves as a test bed for the develop-ment of a next-generation sub-angstrom resolution 3D po-sitioning device. The actuators are10-µm maximum exten-sion piezo stacks from Physik Instrumente [6]. In the courseof the development of the 3D positioning device we have

692

integrated the prototype stage in a digital control environ-ment consisting of dSPACE [7] signal acquisition, signalgeneration, and signal processing modules. The dSPACEhardware is programmed from within MATLAB [8] usingthe simulink graphical programming environment. Since thehardware containing the capacitive sensors has not been cal-ibrated, the gain factor between the input voltage applied tothe piezo and the resulting output voltage of the capacitiveposition sensors is only known approximately and is of minorimportance for the presented results. We will therefore dis-play our measurement results by the internal representationwithin the measurement program in units ofV for the inputsignal, representing the voltage applied to the piezo and alsoin units ofV of the output signal representing the measureddisplacement detected by the sensors. In order to convert thedisplayed values of the input signal to the real voltage on thepiezo, multiply by350, and to convert the output signal to theapproximate displacement innm, multiply by 5000.

The test signal to induce relaxation of the piezo for var-ious offset voltages is schematically shown in Fig. 1 andconsists of a slowly varying sinusoidal offset with relaxationsignals in between. The sinusoid for both the main waveformand the relaxation signal are of sufficiently low frequency(6 10 Hz) so as not to invoke the system dynamics. The sys-tem dynamics can result in overshoot and oscillatory behaviorof the piezo elongation which can significantly change therelaxation behavior of the piezo [9]. In order to reach therelaxed state in a controlled fashion, the amplitude of the re-laxation signal is slowly increased from zero to a maximumvalue and then slowly decreased to zero again. For every off-set value we have measured the sensor response just beforeand just after the application of the relaxation signal.

This procedure has been performed for different valuesof the maximum relaxation amplitudeA, i.e. 0 V, 0.2 V, and0.4 V, see Fig. 2. The durationT of the relaxation signal was30 s for all measurements. The curves show the response ofthe position sensor as a function of the input voltage. Al-though some relaxation is visible, the amount of hysteresis,expressed either in terms of the maximum vertical aperture

INP

UT

VO

LTA

GE

TIME

T

A

Fig. 1. The waveform of the input signal for the relaxation experiments.A slowly varying sinusoid is periodically interrupted by an amplitude-modulated sinusoid of higher frequency for a durationT with a maximumamplitudeA

INPUT [V]

OU

TP

UT

[V]

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.4 V

0.2 V

0 V

Fig. 2. The resulting hysteresis loops for different amplitudes (0 V, 0.2 V,and0.4 V) of the relaxation signal

or the total area enclosed by the loop, has not changed sig-nificantly. Therefore, relaxation does not reduce the hystereticbehavior of the piezo and an anhysteretic curve is not ob-tained using this type of relaxation in piezoelectric materials.Note that the observed tilt of the loops upon application ofthe relaxation signal actually means a change in the effect-ive piezo sensitivity. The effective piezo sensitivity, as definedby the slope of the line connecting the turning points of theloop, increases about35% for the0.4 V relaxation curve ascompared to the0 V curve.

The actual amount of relaxation at every point along theloop for the various relaxation amplitudes is plotted in Fig. 3.For comparison, the input sinusoid (solid line, not to scale)has been added to the graph (Note that the first quarter of thefirst sinusoid corresponds to the virgin curve). The amountof relaxation is periodic with the same period as the inputsignal but not in phase with the input signal. Although, the re-laxation increases with increasing amplitude of the relaxationsignal, the relation is non-linear and has substantial values

INDEX

RE

LA

XA

TIO

N[V

]

0 20 40 60 80 100 120 140-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.4 V

0.2 V

0 V

Fig. 3. The amount of relaxation at every point along the hysteresis loop fordifferent amplitudes (0 V, 0.2 V, and 0.4 V) of the relaxation signal. Thesolid curverepresents the input sinusoid (not to scale)

693

even when the relaxation amplitude is zero. At the extremumvalues of the input signal, the amount of relaxation is equalfor all relaxation amplitudes. Furthermore, for values of theinput signal just beyond the extremum value, the relaxationfor all three measurements becomes zero. We have associatedthe relaxation in the absence of an active relaxation signal, the0 V curve in Fig. 3, with piezo creep. Using this interpreta-tion we observe two distinct locations on the hysteresis loopwhere the creep in the previous experiment becomes zero.

In order to study this effect in more detail, the previousexperiment was repeated but now forA= 0 and for variousinterval timesT, which we will refer to as the delay time.The resulting hysteresis loops for various delay times (0.1,10, and60 s), Fig. 4, show a similar tilt of the loop for increas-ing delay times as was observed for increasing relaxationamplitudes. The effective piezo sensitivity therefore also in-creases as a result of creep (for longer delay times) but theeffect is smaller, about12%. Observe that the behavior at thebeginning of the virgin curve is identical for all creep meas-urements, Fig. 4. Analysis of the same region in the relaxationmeasurements, Fig. 2, also shows identical behavior.

The amount of creep at every point along the hystere-sis loop is displayed in Fig. 5. Again, the creep is periodicwith the same period as the input signal and out of phase.In contrast to the relaxation results, Fig. 3, all curves showzero creep at the same locations with respect to the input sig-nal, independent of the delay time. Note that the horizontalaxis in Fig. 5 indicates the index of the points on the hystere-sis loops and not the time. The time scale for the individualcurves varies from about2 min for the 0.1-s delay curve toabout2 h for the 60-s delay curve. The curves in Fig. 4 canbe interpreted in various ways: the points before and after thedelay procedure can be looked at separately as points defin-ing a non-relaxed curve and a relaxed curve, respectively.Additionally one can define a curve determined by the linesconnecting relaxed points to non-relaxed points, i.e. the tran-sition from a certain location after the delay time to the nextposition just before the delay procedure. As such, a slope canbe associated at the points defining either curve.

INPUT [V]

OU

TP

UT

[V]

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Fig. 4. The resulting hysteresis loops for various delay times (0.1, 10, and60 s)

INDEX

CR

EE

P[V

]

0 20 40 60 80 100 120 140-0.015

-0.01

-0.005

0

0.005

0.01

0.015

60 s

10 s

0.1 s

Fig. 5. The amount of creep for every point along the hysteresis loop. Thesolid line represents the input sinusoid (not to scale). The delay times are0.1, 10, and60 s, respectively

INDEX

SL

OP

E

0 20 40 60 80 100 120 140-0.5

0

0.5

1

1.5

2

0.1 s

10 s

60 s

Fig. 6. The curve slope at every point along the hysteresis loop defined bythe relaxed points for various delay times. Thesolid line represents theslope at the beginning of the virgin curve

In Fig. 6, the slopes at the individual points for the relaxedloops are displayed along with a solid line representing thevalue of the slope at the beginning of the virgin curve. Com-paring Figs. 5 and 6 observe that the locations of the points ofzero creep coincide with the locations on the hysteresis loopwhere the value of the slope of the curve equals the value ofthe slope at the beginning of the virgin curve.

In order to investigate the influence of the amplitude of theinput signal (for fixed delay time) on the behavior of creep,we have measured the piezo response for various amplitudes.In Fig. 7 several hysteresis loops for different amplitudes ofthe input signal are displayed. The amplitude of the input sig-nal was varied from0.1 V to 0.5 V in steps of0.1 V. Forclarity only three measurements are shown but the behaviorfor all measurements at the beginning of the virgin curve isagain identical. The slope at this point therefore is a char-acteristic property of the piezo that is not influenced by theexperiments we have performed. The amount of creep at indi-

694

INPUT [V]

OU

TP

UT

[V]

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Fig. 7. The resulting hysteresis loops for different values of the amplitudeof the input sinusoid (0.1 V, 0.3 V, and0.5 V) and a delay time of10 s

INDEX

CR

EE

P[V

]

0 20 40 60 80 100 120 140

-0.01

-0.005

0

0.005

0.01

0.1 V

0.2 V

0.3 V

0.4 V

0.5 V

Fig. 8. The amount of creep for every point along the hysteresis loop fordifferent values of the amplitude of the input sinusoid (0.1 V, 0.2 V, 0.3 V,0.4 V, and0.5 V). Thesolid line represents the input signal (not to scale)

vidual points along the curve for all measurements is shownin Fig. 8. Again the creep curves are periodic with the sameperiod as the input signal (solid line in Fig. 8) and have com-mon zero crossings. However, the slope connecting relaxedpoints to adjacent non-relaxed points show a relaxation to-wards the value at the beginning of the virgin curve as indi-cated by the solid line in Fig. 9. This observation is consistentwith the experimental results presented by [10] in which sev-eral small signal hysteresis loops have been superimposed ona large signal hysteresis loop. The slope for the smaller loopswas observed to be equal for all loops independent of the pos-ition on the larger hysteresis loop. The zero crossings in thecreep values, Fig. 8, again coincide with the locations on thehysteresis loop where the local slope equals the slope at thebeginning of the virgin curve.

When all locations of zero creep for the individual meas-urements are plotted in one graph, Fig. 10, a trajectory of zerocreep within the operating space of the piezo can be defined.

INDEX

SL

OP

E

0 20 40 60 80 100 120 140-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1

2

3

Fig. 9. The various slopes at each point along the hysteresis loop. The slopefor the curve through the relaxed points (1) follows the non-linearity of thehysteresis loop. The slope for the lines connecting relaxed points to non-relaxed points (2) is nearly constant and equal to the slope at the beginningof the virgin curve (3)

INPUT [V]

OU

TP

UT

[V]

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Fig. 10.The trajectory of zero creep (solid line) as defined by the measuredpoints of zero creep (black dots) superimposed on the hysteresis loops

Note that this trajectory can not be reached in a single stepbut requires a more sophisticated approach procedure. Nev-ertheless, the trajectory could be very useful in positioningexperiments. The slope relaxation and the relation betweenthe locations of zero creep and the local slope suggest a morefundamental relation between the creep and the slope of thehysteresis loop.

In Fig. 11 a creep experiment is displayed for small indi-vidual steps and sufficient delay time (10 s) between each stepto fully relax (within the resolution of our measurements) atevery position. The inset shows the region around the turningpoint at the upper right corner. Observe that the direction ofthe creep remains upward even after the direction of the inputsignal has changed. Since we use a sinusoidal input signal, thesize of each individual voltage step changes along the curve.In order to compare the creep along the curve we have cal-culated the creep per unit step size for every point along the

695

INPUT [V]

OU

TP

UT

[V]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.97 0.975 0.98 0.985 0.99 0.995 10.86

0.864

0.868

0.872

0.876

0.88

Fig. 11.Creep experiment for the virgin curve and a small section of the sta-tionary loop. The inset, with the details near the turning point, shows thatthe direction of the creep is not determined by the direction of the inputsignal

curve, see Fig. 12. In the same figure, the difference betweenthe local slope at each individual location along the curve andthe slope at the beginning of the virgin curve is displayed. Thestriking equivalence between the two data sets suggest the fol-lowing relation between the local slope and the total amountof creep:

C (∆V)=[

dH

dV− dH

dV

∣∣∣∣V=0

]∆V , (1)

whereC(∆V ) is the total amount of creep,H a pair of func-tions describing the ascending and descending branches ofthe hysteresis loop,dH

dV the local slope,dHdV

∣∣V=0

the slope atthe beginning of the virgin curve, and∆V the step size ofthe input signal. Note that∆V for the upper or descending

INDEX

CR

EE

P/S

TE

PS

IZE

0 20 40 60 80 100 120 140-8

-6

-4

-2

0

2

4

6

8

10

Fig. 12. The amount of creep per unit step size for every point along thecurve (dots) is equal to the difference between the local slope and the slopeat the beginning of the virgin curve (solid line)

branch is negative and that∆V for the lower or ascendingbranch is positive. Observe that, althoughC(∆V ) is propor-tional to∆V, the direction of the creep is not changed when∆V changes sign becausedH

dV − dHdV

∣∣V=0 also changes sign at

the turning points. It is only around the locations where theslope equalsdH

dV

∣∣V=0 that the creep crosses zero and changes

sign. From (1) we can derive that the creep is essentially de-scribed by the local non-linearity or deviation of the localslope as compared to the slope at the beginning of the virgincurve.

2 Reduction of hysteresis and creep

The experimental observation (1) would suggest that thecreep can be minimised if the deviations of the slope of thetransfer functionH can be minimised. In other words, thecreep can be minimised if the linearity of the transfer canbe optimised. This can be achieved by, for example, reduc-ing the hysteresis in the system. This hypothesis can be testedon piezoelectric systems that exhibit no hysteresis, as forexample quartz. However, quartz is a very insensitive piezo-electric material and requires large voltages to produce a sig-nificant displacement. Furthermore, our set-up does not allowan easy exchange of the actuators. Another way to reduce thehysteresis in ceramic piezoelectric actuators can be realisedby using charge control instead of voltage control. Since pre-vious papers [11–13] have reported on a reduced hystereticbehavior when the piezo is driven with a charge instead ofwhen driven with a voltage, a test set-up has been realized toassess the possibilities of charge control. A voltage-to-chargeconverter has been realized using a single opamp [12–14]as shown in Fig. 13a. Although the maximum output volt-age swing is restricted to the range of the supply voltage(+/– 15 V) it was verified that sufficient hysteresis could beobserved in the case of voltage control. The FET input opampTL071 has been selected for its relatively low bias current(Ib = 200 pA) and availability. Since all measurements wereperformed in the quasi-static regime, the dynamic behavior ofthe opamp is of little concern. The external capacitor was se-lected for low leakage. Analysis of the circuit yields, for anideal opamp and an ideal external capacitor:

U+ =U− and Ib = 0 ,

so

Uin =UC ,

and

Qext=UC×Cext=Uin×Cext .

+

-

Uin

Cext

Piezo-15 V

+15 V

Uout

UcIb

+

-

Uin

Piezo

-15 V

+15 V

Uout

a b

Fig. 13a,b.Electronic circuits for charge control (a) and voltage control (b)

696

Since

Qext= Qpiezo ,

we have

Qpiezo=Uin×Cext , (2)

that is, the charge on the piezo is proportional to the inputvoltage with a conversion factor determined by the value ofthe external capacitor, irrespective of the piezo capacitance.

In order to exclude any influence from the opamp incomparing the charge control measurement and the voltagecontrol measurement the same opamp has been used in a cir-cuit for voltage control, see Fig. 13b. Following similar argu-ments, analysis of the circuit yields

Uin =U+ =U− =Uout , (3)

so the voltage across the piezo equals the input voltage.In Fig. 14 the voltage and charge measurements are dis-

played in the same graph. In contrast to the previous experi-ments, the waveform of the input signal was triangular inorder to make the voltage steps equidistant. For clarity, thecurves have been separated by an artificial offset. Since thetransfer between the input voltage and the resulting elonga-tion of the piezo is different for the two electronic circuits(2),(3), we have adjusted the input voltage for both measure-ments so as to obtain approximately the same piezo response.Additionally, the input voltage used in the charge control ex-periment has been scaled in Fig. 14 to the values used in thevoltage control experiment. This way we are able to comparethe creep for voltage and charge control experiments. Becauseof the simplicity of the electronic circuit we used to performcharge control, significant leakage currents were present re-sulting in drift in the charge control measurement. The chargecontrol curve in Fig. 14 has been corrected for linear drift.Compared to the voltage controlled piezo, the hysteresis in

INPUT [V]

OU

TP

UT

[V]

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

CHARGE CONTROL

VOLTAGE CONTROL

Fig. 14.Creep experiments in a system with reduced hysteresis for a charge-controlled piezo compared to a voltage-controlled piezo. In order to com-pare the two measurements the values of the input voltage for chargecontrol have been scaled to the values used in the voltage control experi-ment

Fig. 15. Creep at individual locations along the hysteresisloops for chargecontrol and voltage control. The triangular input signal (solid line) is not toscale

INDEX

CR

EE

P/S

TE

PS

IZE

0 10 20 30 40 50 60 70-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Fig. 16. Comparison between the creep per unit step size (solid line) andthe difference between the local slope and the slope at the beginning ofthe virgin curve (dotted line) for the charge control measurement. Althoughthe data contain more noise as compared to previous results, the correlationbetween the two curves suggests that equation (1) is still valid

the charge control experiment has been reduced to about 1/3of the value measured in the voltage control experiment.

The creep at individual points for both experiments is dis-played in Fig. 15 along with the triangular input signal. Theamount of creep in the charge control measurement is sig-nificantly reduced as compared to the voltage control meas-urement. Furthermore, the phase of the creep in the chargecontrol measurement is almost inverted as compared to thephase of the creep in the voltage control signal.

Figure 16 furthermore indicates that the amount of creepfor the charge control measurement is also determined by (1).More detailed inspection of the charge control measurementreveals a residual non-linearity, that suggests that even forzero hysteresis, the transfer will still be non-linear and thecreep will not completely vanish.

697

3 Conclusions

The relaxation mechanism in piezoelectric materials does notresult in an anhysteretic curve and also does not reduce thehysteresis. The effective sensitivity of the piezo is increasedupon application of relaxation signals (up to35%) and as a re-sult of creep (up to12%). This effect may be important inthe calibration procedures of (open loop) piezo materials. Es-pecially in scanning probe applications, a distinction can bemade between a fast scanning direction and a slow scanningdirection resulting in two different times scales for the scan-ning directions. Even when identical piezoelectric actuatorsare used, the effective piezo sensitivity will be different foreach direction.

Several points on the hysteresis loop exhibit zero relax-ation and zero creep. Within the operating voltage of thepiezo, the points of zero creep form a trajectory of zero creepthat may be useful in positioning strategies. The slope at thesepoints equals the slope at the begining of the virgin curve.Once the virgin curve of the piezo is characterised, the pos-itions of zero creep can be obtained in closed-loop controlsystems by measuring the local slope. In contrast to com-mon belief, the direction of the creep is not primarily deter-mined by the direction of the input signal but solely by theshape of the hysteresis loop. From the observation of pointsof zero creep we have found a more general relation betweenthe amount of creep and the slope of the hysteresis loop in

piezoelectric actuators. Finally, the assumption imposed bythe experimentally obtained relation between the creep andthe local slope, that the creep should reduce when the linear-ity is improved has been verified on a system with reducedhysteresis.

Acknowledgements.This research is supported by the Technology Founda-tion (STW).

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